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NSAA 2022 Mathematics PART A

20 questions20 marksUpdated October 2025

The NSAA 2022 Mathematics PART A paper in full: all 20 questions, each with its answer. NSAA is the Natural Sciences Admissions Assessment. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

Which one of the following is a simplification of

y(3x12zy3)2y {\left( \frac{3x^{\frac{1}{2}}z}{y^3} \right)}^2
  • A.3xz2y4\frac{3xz^2}{y^4}
  • B.3xz2y5\frac{3xz^2}{y^5}
  • C.9x2z2y5\frac{9x^2z^2}{y^5}
  • D.9xz2y4\frac{9xz^2}{y^4}
  • E.9xz2y5\frac{9xz^2}{y^5}
  • F.59x2z2y5\frac{5}{9x^2z^2y^5}

Answer: E

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Question 2

Triangle PQR has a right angle at Q.

The point T lies on QR such that QT=14QRQT = \frac{1}{4}QR.

PT=6PT = 6 cm

PR=12PR = 12 cm

What is the length of QT, in cm?
  • A.2
  • B.232\sqrt{3}
  • C.322\frac{3\sqrt{2}}{2}
  • D.655\frac{6}{5}\sqrt{5}
  • E.2721\frac{2}{7}\sqrt{21}

Answer: D

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Question 3

Find the complete set of values of xx that satisfy the inequality

34(6−x)−12(6−x)−x<0\frac{3}{4}(6-x) - \frac{1}{2}(6-x) - x < 0
  • A.x<−13x < -\frac{1}{3}
  • B.x>−13x > -\frac{1}{3}
  • C.x<35x < \frac{3}{5}
  • D.x>35x > \frac{3}{5}
  • E.x<−34x < -\frac{3}{4}
  • F.x>34x > \frac{3}{4}
  • G.x<−32x < -\frac{3}{2}
  • H.x>−32x > -\frac{3}{2}

Answer: D

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Question 4

I have two fair dice, X and Y, each of which has six sides.

The faces on X are labelled 1, 1, 2, 3, 4, 5.

The faces on Y are labelled 2, 3, 4, 5, 6, 6.

I roll the dice together and calculate my total score by adding the number rolled on X to the number rolled on Y.

What is the probability that my total score is greater than 9?
  • A.14\frac{1}{4}
  • B.16\frac{1}{6}
  • C.19\frac{1}{9}
  • D.512\frac{5}{12}
  • E.536\frac{5}{36}

Answer: E

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Question 5

Rob keeps a record of what he earns each day.

On Monday, he earned 50% less than he earned on Sunday.
On Tuesday, he earned 20% more than he earned on Monday.
On Wednesday, he earned 30% less than he earned on Tuesday.

On Wednesday, he earned £84.

How much did Rob earn on Sunday?
  • A.£15.12
  • B.£35.28
  • C.£117.60
  • D.£200
  • E.£210
  • F.£300
  • G.£1200

Answer: D

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Question 6

The nthn^{th} term of a sequence T is (n−3)2(n - 3)^2, where nn is a positive integer.

The nthn^{th} term of another sequence V is 3n+p3n + p, where pp is a constant and nn is a positive integer.

The 10th10^{th} term in T is equal to twice the 5th5^{th} term in V.

What is the 4th4^{th} term in V?
  • A.-16
  • B.4
  • C.16.5
  • D.21.5
  • E.31
  • F.46
  • G.95

Answer: D

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Question 7

Which one of the following is a simplification of

5x2−17x−1225x2−9÷x2+x−12x2−x−6\frac{5x^2-17x-12}{25x^2 - 9} \div \frac{x^2 + x -12}{x^2-x-6}
  • A.(x−4)(x+2)(x−3)(x+4)\frac{(x-4)(x+2)}{(x-3)(x+4)}
  • B.(x−3)(x+2)(5x−3)(x+3)\frac{(x-3)(x+2)}{(5x-3)(x+3)}
  • C.(x−4)(x+2)(5x−3)(x+4)\frac{(x-4)(x+2)}{(5x-3)(x+4)}
  • D.(x−4)(x−3)(5x−3)(x−6)\frac{(x-4)(x-3)}{(5x-3)(x-6)}
  • E.(x+2)(5x+3)\frac{(x+2)}{(5x+3)}
  • F.(x+4)(x−6)(5x+3)(x+2)\frac{(x+4)(x-6)}{(5x+3)(x+2)}
  • G.(x−3)(x+2)(5x+3)(x+3)\frac{(x-3)(x+2)}{(5x+3)(x+3)}

Answer: C

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Question 8

S is a list of six numbers:

1,2,x,x+1,x+1,151, 2, x, x + 1, x + 1, 15 where 2≤x≤142 \le x \le 14

The mean of S is one more than the median of S.

What is the value of xx?
  • A.2232\frac{2}{3}
  • B.3233\frac{2}{3}
  • C.4234\frac{2}{3}
  • D.5235\frac{2}{3}
  • E.6236\frac{2}{3}

Answer: B

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Question 9

A rectangle PQRS has length (2x−1)(2x - 1) cm and width (x+1)(x + 1) cm as shown on the diagram.

A larger rectangle is made by adding 3 cm to both the length and the width of PQRS, as shown.

The larger rectangle has an area of 360 cm2^2.

Exam diagram


[diagram not to scale]

What is the ratio of PQ to PS?
  • A.1:2
  • B.4:7
  • C.5:8
  • D.7:11
  • E.10:17
  • F.17:31

Answer: B

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Question 10

tt is inversely proportional to the square of ww.

tt and ww are positive numbers.

t=36t = 36 when w=2×10−2w = 2 \times 10^{-2}

What is the value of ww when t=100t = 100?
  • A.1.2×10−41.2 \times 10^{-4}
  • B.1.2×10−21.2 \times 10^{-2}
  • C.1.44×10−61.44 \times 10^{-6}
  • D.1.44×10−31.44 \times 10^{-3}
  • E.103×10−4\frac{10}{3} \times 10^{-4}
  • F.103×10−2\frac{10}{3} \times 10^{-2}
  • G.7.2×10−67.2 \times 10^{-6}
  • H.7.2×10−37.2 \times 10^{-3}

Answer: B

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Question 11

PQRS is a trapezium as shown.

Exam diagram


tan⁡RSQ=58\tan RSQ = \frac{5}{8}

What is the length of PS, in metres?
  • A.45
  • B.65
  • C.80
  • D.120
  • E.25+403325 + \frac{40\sqrt{3}}{3}
  • F.40+643340 + \frac{64\sqrt{3}}{3}
  • G.25+40325 + 40\sqrt{3}
  • H.64+40364 + 40\sqrt{3}

Answer: G

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Question 12

A cyclist rides along a track to the top of a hill then immediately turns around and descends along the same track to her starting point.

She takes 40 minutes at an average speed of 12 kmh−112 \text{ kmh}^{-1} to reach the top.

Her average speed for the whole journey is 15 kmh−115 \text{ kmh}^{-1}.

What is the average speed of her descent?
  • A.16 kmh−116 \text{ kmh}^{-1}
  • B.18 kmh−118 \text{ kmh}^{-1}
  • C.20 kmh−120 \text{ kmh}^{-1}
  • D.24 kmh−124 \text{ kmh}^{-1}
  • E.30 kmh−130 \text{ kmh}^{-1}

Answer: C

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Question 13

A solid cylinder has radius rr cm and height hh cm.

A cube has side length 3r3r cm.

The total surface area of the cylinder is equal to four times the total surface area of the cube.

Which of the following is an expression for hh in terms of rr?
  • A.(18π−2)r\left(\frac{18}{\pi} - 2\right)r
  • B.(18π−1)r\left(\frac{18}{\pi} - 1\right)r
  • C.27rπ\frac{27r}{\pi}
  • D.(27π−1)r\left(\frac{27}{\pi} - 1\right)r
  • E.(274π−1)r\left(\frac{27}{4\pi} - 1\right)r
  • F.108rπ\frac{108r}{\pi}
  • G.(108π−1)r\left(\frac{108}{\pi} - 1\right)r
  • H.(108π−2)r\left(\frac{108}{\pi} - 2\right)r

Answer: G

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Question 14

Consider the equation 2x2+4x+c=02x^2 + 4x + c = 0, where cc is a constant.

The positive difference between the roots of this equation is 10\sqrt{10}.

What is the value of cc?
  • A.-5
  • B.-4.5
  • C.-3
  • D.-0.5
  • E.0.75
  • F.8

Answer: C

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Question 15

The variables xx and yy are related by the equation:

x=5−2y3+11−2y3x = 5 - \frac{2y^3 + 1}{1 - 2y^3}


Which of the following is a rearrangement to make yy the subject?
  • A.y=x−48x−483y = \sqrt[3]{\frac{x-4}{8x-48}}
  • B.y=x−68x−323y = \sqrt[3]{\frac{x-6}{8x-32}}
  • C.y=x−2x−63y = \sqrt[3]{\frac{x-2}{x-6}}
  • D.y=x−3x−43y = \sqrt[3]{\frac{x-3}{x-4}}
  • E.y=x−42x−123y = \sqrt[3]{\frac{x-4}{2x-12}}
  • F.y=x−62x−83y = \sqrt[3]{\frac{x-6}{2x-8}}

Answer: E

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Question 16

PQR is a triangle as shown.

S and T are points on the sides PQ and PR.

ST is parallel to QR.

PS = 3 cm
ST = x cm
QS = (x + 2) cm
QR = (2x + 1) cm

Exam diagram


[diagram not to scale]

What is the length, in cm, of QR?
  • A.2+52+\sqrt{5}
  • B.2+132+\sqrt{13}
  • C.5+275+2\sqrt{7}
  • D.5+2115+2\sqrt{11}
  • E.7
  • F.9

Answer: B

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Question 17

Three different numbers are chosen at random from 1,2,3,4,5\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}.

What is the probability that the three numbers form the three sides of a right-angled triangle?
  • A.115\frac{1}{15}
  • B.110\frac{1}{10}
  • C.310\frac{3}{10}
  • D.13\frac{1}{3}
  • E.25\frac{2}{5}
  • F.23\frac{2}{3}
  • G.45\frac{4}{5}

Answer: E

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Question 18

P, Q and R are regular polygons.

Q has three times as many sides as P.

An interior angle of Q is 10∘10^\circ larger than an interior angle of P.

R has twice as many sides as Q.

How much larger is an interior angle of R than an interior angle of Q, in degrees?
  • A.2122\frac{1}{2}
  • B.5
  • C.6236\frac{2}{3}
  • D.7127\frac{1}{2}
  • E.10
  • F.15
  • G.162316\frac{2}{3}

Answer: A

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Question 19

The point (−1,5)(-1, 5) is translated to the point (3,2)(3, 2) by two successive translations.

The first translation is by the vector (3p −4p)\begin{pmatrix} 3p \ -4p \end{pmatrix}.

The second translation is by the vector (q −2q)\begin{pmatrix} q \ -2q \end{pmatrix}.

What is the value of p+qp + q?
  • A.-14
  • B.-7
  • C.-5
  • D.-1
  • E.1
  • F.5
  • G.7
  • H.14

Answer: D

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Question 20

Consider the graphs of the form

y=x2+2ax+ay = x^2 + 2ax + a


What is the complete range of values of aa for which the minimum point of the graph lies above the x-axis?
  • A.There are no values of aa
  • B.a<0a < 0
  • C.0<a<10 < a < 1
  • D.−1<a<1-1 < a < 1
  • E.a<−1a < -1 or a>1a > 1
  • F.a<0a < 0 or a>1a > 1
  • G.aa can take any value

Answer: C

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NSAA 2022 Mathematics PART A: Questions & Worked Solutions | esat.fyi